So the object Ept:=[I,B]×Bpt\mathbf{E}_{pt} := [I,B]\times_{B} pt is defined to be the pullback of the diagram [I,B]→d1B←ptBpt [I,B] \stackrel{d_1}{\to} B \stackrel{pt_B}{\leftarrow} pt and the morphism EptB→B\mathbf{E}_{pt}B \to B is the composite of the left vertical morphism in the above diagram which comes from the definition of pullback and d0d_0.

Then a (generalized) “BB-bundle” on some object XX is a morphism P→XP \to X which is the pullback of the generalized universal BB-bundle Ept\mathbf{E}_{pt} along a “classifying morphism” g:X→Bg : X \to B

If one defines, as one does, a (possiby directed) homotopy between two morphisms f,g:A→Bf,g : A \to B to be a morphism η:A→[I,B]\eta : A \to [I,B] such that d0*η=fd_0^* \eta = f and d1*η=gd_1^* \eta = g, then PP is the “lax pullback” (really comma object) of the point along gg

is exact in that ii is the kernel of pp in the sense of kernels of morphisms of pointed objects (see there).

Examples

Groupoid incarnations of universal principal bundles

In (higher) categorical contexts, take the interval object to the the interval category I:={a→b}I := \{a \to b\}. Then

Ordinary GG-principal bundles

For C=C =Cat, B:=BGB := \mathbf{B}G a one-object groupoid corresponding to a group GG with the unique point, EptBG=EG=G//G\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G = G//G is the action groupoid of GG acting on itself. The sequence of groupoids is

G→G//G→BG.
G \to G//G \to \mathbf{B}G
\,.

This is the universal GG-bundle in its groupoid incarnation. It is a theorem by Segal from the 1960s that indeed this maps, under geometric realization to the familiar universal GG-bundle in TopTop. Moreover, it can be seen that every GG-principal bundle P→XP \to X in the ordinary sense is the pullback of EG\mathbf{E} G in the following sense:

the GG-bundle P→XP \to X is classified by a nonabelian GG-valued 1-cocycle (the transition function of any of its local trivializations), which is an anafunctor

The universal groupoid bundle EG→BG\mathbf{E}G \to \mathbf{B}G may now be pulled back along this anafunctor to yield the groupoid bundle g*EG→Xg^* \mathbf{E}G \to X given by the total left vertical morphism in

GG-principal 2-bundles

For C=2CatC = 2Cat, strict 2-categories , B:=BGB := \mathbf{B}G a strict one-object 2-groupoid corresponding to a strict 2-groupGG with the unique point, EptBG=EG\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G was described under the name INN(G)INN(G) in

Action groupoids as generalized bundles

A morphism ρ:B→F\rho : B \to F to a pointed objectFF (needs not be a basepoint preserving morphism!) can be regarded as a representation of BB on the point of FF. The pullback of the universal FF-bundle along this morphism

ρ*EptF→B \rho^* \mathbf{E}_{pt} F \to B

can be addressed as the FF-bundle ρ\rho-associated to the universal BB-bundle EptB\mathbf{E}_{pt}B.

If BB is a groupoid, then ρ*EptF\rho^* \mathbf{E}_{pt} F is the action groupoid of BB acting on the point of FF.

Further pulling this back along a cocycle g:X^→Bg : \hat X \to B of a BB-principal bundle yields the ρ\rho-accociated bundle of that.

For instance for B=BGB = \mathbf{B}G and F=VectF = Vect with ρ:BG→Vect\rho : \mathbf{B}G \to Vect a representation of the group GG on a vector space VV, the ρ\rho-associated Vect\mathrm{Vect}-bundle on BG\mathbf{B}G is

V→V//G→BG.
V \to V//G \to \mathbf{B}G
\,.

Pulling that further back along the cocycle g:X^→BGg : \hat X \to \mathbf{B}G classifying a GG-principal bundle P→XP \to X, one obtains the familiar vector bundle P×GV→XP \times_G V \to X which is ρ\rho-associated to PP, along the lines described above: